In certain areas of medical imaging, such as transmission CT of the beating heart and emission CT using multiwire proportional chambers in nuclear medicine, data collection over a complete range of views is impossible. We propose to investigate mathematical solutions to the problem of image reconstruction from a limited range of views (together with suitable numerical methods for the computer implementation of the solutions). The aim is to devise and test new procedures based (i) on an analytic solution, (ii) on extrapolation of projection data using gradient information, and (iii) on combinations of such methods. The performance of various algorithms will be compared using both computer-generated projections of realistic mathematical phantoms and actual clinical data.